In this paper, for the sake of reducing the training complexity of the additional NN FN, we remove the max-pooling layer and simplify the CNN to have only two convolutional layers and one dense layer. The convolutional layer explores the features of the NN FN input message \(u_i(x_i)\) .
The dense layer is appended to process the features and then delivers the output to the VNs.
As shown in Fig. REF , the first convolutional layer has \(f_{n_1}\)  filters and each filter has a size of \( f_{l_1}\) , where \(f_{l_1}\)  indicates that \(f_{l_1}\)  messages are considered by the filter during each convolutional step.
Let \(f_{s_1}\)  denote the stride of the sliding window in the first convolutional layer.
With the attachment of zero padding, the output dimension of the first convolutional layer reduces to \( \lceil N/f_{s_1}\rceil \) .
The filter convolves with the messages from \(u_1(x_1)\)  to \(u_N(x_N)\)  to compensate any detrimental effect in the message passing that may degrade the detection performance.
A similar operation of filtering is performed in the second convolutional layer which has a corresponding hyper-parameter set of \((f_{n_2}, f_{l_2}, f_{s_2})\) .
The output of the convolutional layer is sent to the dense layer to combine and process the filters' results before delivering the messages to the VNs.
The dense layer processes all the filters' results and outputs the message \(v_i(x_i)\)  for \(i \in \lbrace 1,...,N\rbrace \) .
The negative of a rectified linear activation function (ReLU) is used as the activation function in the dense layer [1]}. During each iteration, the CNN has initial weights and biases randomly generated from a truncated normal distribution with a standard deviation of \(\sigma _{\text{CNN}_1}\)  and \(\sigma _{\text{CNN}_2}\)  for the first and second CNN layers, respectively.
For each iteration, messages \(u_i(x_i)\)  will be updated by using (REF ) then passed to the CNN.
Messages are sent back from the CNN to join the APPs accumulation according to (REF ).
By unfolding the FG including the NN FN to an NN system with multiple layers, the overall detector can be trained and optimized.
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